Questions tagged [ordinary-differential-equations]

For questions about ordinary differential equations, which are differential equations involving ordinary derivatives of one or more dependent variables with respect to a single independent variables. For questions specifically concerning partial differential equations, use the [tag:pde] instead.

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19 views

Flow of a vector field on a manifold with boundary

Let $(M,g)$ be a compact riemannian manifold with boundary, and let $N$ be the unit normal vector field defined on $\partial M$ and pointing outwards. Let $X$ be a smooth vector field defined on $M$, ...
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19 views

Solutions for small perturbations

I have the following exercise: Let $f,g\colon \mathbb{R}\rightarrow \mathbb{R}$ be $C^1$ bounded functions and $f(0)=0$. Show that for suitably small $\varepsilon>0$ the following problem has a ...
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4answers
36 views

Integrating factor mistakes when solving 1 order ODE

I have an ODE: $$\frac{dy}{dx} + 3x^{2}y = x^{2}$$ . I got the following integrating factor: $$e^{x^3}$$ Then I multiplied both sides, but didn't come up with the right answer. It should be: $$...
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18 views

Differential equation of a RRC circuit as Voltage output/voltage input?

So in this presentation i have got, there is a RRC circuit, and they get this differential equation, but there is no procedure explained how they have got it. This is basicially a series RRC circuit ...
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0answers
25 views

Logistic Equation Solution - Nurgaliev's law

I am trying to solve dN/dt = bN^2 - dN Please see the image of my full working. Working out I feel like I am messing up on the partial fractions part, because my final solution converges towards ...
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1answer
15 views

Jacobian of RHS of ODE non-square matrix

I'm asked to convert the following ODE into a system of order $1$: $$u''(t) = f(t,u(t),u'(t))$$ and I came up with: $$\begin{align*}u_0'(t) &= u_1(t)\\u_1'(t) &= f(t,u_0(t),u_1(t))\end{...
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17 views

Compute approximation of ODE using one step of explicit/implicit Euler method

I'm given the IVP: $$u^{(3)}(t) + u'(t) = tu(t)$$ $$u''(2) = 2$$ $$u(2) = 0$$ $$u'(2) = 1$$ and am asked to approximate the solution for $t=2.5$ using one step of the explicit Euler-method and one ...
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1answer
34 views

Is it possible for the function to be a solution of this equation?

Consider the equation $$y′′ + p(t)y′ + q(t)y^3 = 0$$ with $p, q ∈ C(\mathbb{R}, \mathbb{R})$. Is it possible for the function $$y = e^t −\frac{t^2}{2} −t−1$$ to be a solution of this equation? I ...
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3answers
60 views

Differential Equation, how to solve $(x^2-2xy-y^2)dy= (x^2+2xy-y^2) dx​$?

Solve$$(x^2-2xy-y^2)dy= (x^2+2xy-y^2) dx​$$ I've tried many methods but still unable to solve that problem
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1answer
32 views

Show that $y(x)= \int_x^\infty e^{-t^2}dt$ satisfies the differential equation $y^{(2)}+2xy^{(1)}=0$. [on hold]

Show that $y(x)= \int_x^\infty e^{-t^2}dt$ satisfies the differential equation $y^{(2)}+2xy^{(1)}=0$.
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1answer
26 views

$x(t)$ is a solution of the equation $x′ = f(t,x)$ such that $x(0) \neq 0$, show that $x(t) \neq 0,~ \forall t ∈ \mathbb{R}$.

Let's consider the following problem. Let $f(t,x) ∈ C(\mathbb{R} × \mathbb{R}^n,\mathbb{R}^n)$ and satisfy a local Lipschitz condition in $x$. Assume $f(t,0) = 0$. If $x(t)$ is a solution of the ...
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2answers
22 views

Finding constant $a$ so that differential equation becomes correct

Question: Find the value of $a$ so that the function $$y = \sqrt{x} \ln{x}$$ is a solution to the differential equation $$y' - \frac{a}{x} \cdot y = \frac{1}{\sqrt{x}}$$ Attempted solution: My ...
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3answers
64 views

Why Special Functions are called 'special'?

Why Special Functions are called 'special' ? What particular thing made it so special ?
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1answer
36 views

Why the origin is globally asymptotically stable?

If the Lyapunov function is $$ V(x) = x^2_1 + x^2_2-1 $$ And its time derivative is $$ \dot{V}(x) = -(x^2_1 + x^2_2) $$ Why the origin is globally asymptotically stable?
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2answers
43 views

Difficult Second Order ODE solution?

I'm looking at the functional $$T[y] = \int_a^b\sqrt{\frac{1+y'^2}{2g(y-\mu x)}}\ \mathrm dx$$ and trying to minimize it via the Euler-Lagrange-equations, which I can do, however, it seems that the ...
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9answers
1k views

How do we know $\sin$ and $\cos$ are the only solutions to $y'' = -y$?

According to Wikipedia, one way of defining the sine and cosine functions is as the solutions to the differential equation $y'' = -y$. How do we know that sin and cos (and linear combinations of them,...
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0answers
14 views

A piecewise function as the output signal of an LTI system

In an LTI system, consider the following: The input signal: $$ x(t)= \begin{cases} 16 \quad & ; -7<t<0 \\ 0 & ; \text{otherwise} \\ \end{cases} $$ And the unit impulse ...
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2answers
43 views

Solution to $\frac{d^n y}{dx^n} = y$ for integer $n$

In short, my question is about the solution of $\frac{d^n y}{dx^n} = y$ for positive integer $n$, where $y$ is real. For example, if $n = 2$, the solution is $y = c_1e^x+c_2e^{-x}$. The work that I ...
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22 views

Obtaining function from integrals

I have a question I cannot figure out if it's correct or not. I have a function $f(x,y)$ unknown which I'd like to obtain. Starting from $q(y) = \int dxf(x,y)$ and $g(x) = \int dyf(x,y)$ I think I ...
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1answer
23 views

Differential equation of a LRC circuit as Voltage output/voltage input?

So in this presentation i have got, there is a LRC circuit, and they get this differential equation, but there is no procedure explained how they have got it. This is basicially a series LRC circuit ...
1
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2answers
47 views

Finding the solution to $xy'' +2y' +xy=0$ around $x_{0}=0$using the method of Frobenius.

We know that the solution of this ODE is like: $$ y=\sum_{n=0}^{\infty}C_nx^{n+r}$$ Them derivative $y$ and $y'$. $$y'=\sum_{n=0}^{\infty}(n+r)C_nx^{n+r-1}$$ $$y''=\sum_{n=0}^{\infty}(n+r-1)(n+r)C_nx^...
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2answers
36 views

Which method to solve this differential equation?

$$x'=(x+t+1)^2$$ I need to solve this differential equation but do not know how. We cannot use separation of variables so my only guess here would be to use an integrating factor but how would I find ...
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22 views

Differential equation of Serial RL which are connected to parallel C

I have a serial RL which are connected to a capacitor in parallel (Here is the circuit), I'm trying to find the governing differential equation od teh circuit. So far, I have tried using Kirchhoff's ...
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1answer
56 views

General solution to $\frac{d^{2}y}{dt^{2}}+f(t)y=0$

Is there a simple method to find the general solution to the ODE: $$\frac{d^{2}y}{dt^{2}}+f(t)y=0\ .$$ Assume all the usual nice things about $f(t)$ like continuity and smoothness.
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54 views

Differential Equation $ \frac{dy}{dx}=\tan(\sqrt{y^2 + x^2}) $

$ \frac{dy}{dx}=\tan(\sqrt{y^2 + x^2}) $ I can't solve this ODE, no Idea on what to do, posted yesterday one similar, however this is the one answering to a problem
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36 views

Does this IVP have a unique solution?

I have looked at some simpler examples on ODEs, but for this problem the $e^x$ part confuses me. Original Problem: Does this IVP have a unique solution? $$ y' = e^x + (x/y), ~~~ y(0) = 1 $$ Can ...
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1answer
14 views

Determining accretion rate of a cube.

I have a cube of size $h$, which is 'accreting' at a fixed rate $R$ - i.e. volume ($V$) is added in proportion to the surface area ($A$) of the cube, where $R$ is in units of $\frac{V}{A t}$. I know ...
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32 views

How to solve the smallest eigenvalue of a 1st-order linear ODE system?

I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities \begin{align} -\mathrm{i} u'(x) +f^*(x) ...
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1answer
37 views

Is it possible to solve this second order autonomous differential equation?

$$x''(t) = \frac{1}{x^2(t)}$$ I'm interested in this differential equation because it mimics the motion of an object subject to gravity. The solution of this differential equation will be an ...
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1answer
34 views

How can I estimate this second order differential equation?. $ \ ((y'(x))^2+1)^{3/2}\cos(x) = y''(x) \ \ ,y(0)=0, y'(0)=1 \ $

$$ \ ((y'(x))^2+1)^{3/2}\cos(x) = y''(x) \ \ ,y(0)=0, y'(0)=1 \ $$ where do I start?. I know using the Maclaurin series. what should I use an infinite product or infinite sum. it doesn't work well.
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2answers
54 views

Different ODE: tan(y)=y'

I made up a problem where this ODE appears and is the solution to that, well I'm supposed to say what I've tried but I'm not even at university and never even taken calculus class. However I'm a bit ...
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1answer
27 views

Explicit Solution to Euler Lagrange Equation of Shortest Distance between two Points

So if $f(t)=(x(t),y(t))$ with $f(0)=a$ and $f(1)=b$, I should minimize $L(f)=\int_0^1{|\dot{f(t)}|}dt$. I get a jumble of equations when solving the Euler Lagrange equations with respect to $t$. Would ...
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1answer
30 views

Unit impulse response of a discrete-time LTI system

The problem: Consider a discrete-time LTI system. If the output signal is: $$ y[n]=5 \left( \frac{1}{5} \right) ^n u[n] -2^{-n} u[n] $$ , then the input signal will be: $$ x[n]=\left( \frac{...
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0answers
25 views

Expressing Bessel's Function at $\nu = \pm \frac{1}{2} $ using $\sin(z)$ and $\cos(z)$

I'm trying to solve this question: We can prove that $u$ is a solution of Bessel's Equation $z^2 u'' + zu + (z^2 -\nu^2) = 0 $ if and only if $w(z) = z^{\frac{1}{2}}u(z)$ is a solution of $$ z^2 w''...
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Reference for Engineering Mathematics on web

i've a course at the college that this is content Complex algebra and its geometrical interpretation, Multivariate real valued functions and their taylor expansion. Elements of vector analysis (...
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1answer
35 views

stability of dynamic system, find control

Our system is $\dot{w} = Aw+Bv$ where $A = \begin{pmatrix}1 & 0 & 2 \\ 1 & 7 & 3\\1 & 2 & 0\end{pmatrix}, B = \begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix}$ and our goal is to ...
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1answer
35 views

Where can I find this Gronwall's inequality proof?

I'm looking for a proof of the following result: Let I denote an interval of the real line of the form $[a,b]$ with $ a<b $. Let $\beta$ and $u$ be real-valued continuous functions defined on I. ...
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1answer
41 views

Euler (equidimensional) equation question

Consider the equation $$x^2y''-8xy'+20y=0.$$ From an undergraduate ODE course, it is known that the two linearly are $y_1=x^5$ and $y_2=x^4$. However, why don't we consider solutions, for example, ...
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Help with a transport problem regarding finding unsteady concentration profiles [on hold]

A sphere of radius R containing species A at a uniform concentration $C_A0$ is immersed at t = 0 into an infinite bath containing none of species A. Species A diffuses to the surface of the sphere ...
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63 views

Derive PDE for unsteady heat conduction in a cylinder [on hold]

A long cylinder of radius R and thermal conductivity $\alpha$ is initially at temperature $T_0$. Suddenly the cylinder is dropped into a constant temperature bath maintained at temperature $T_1$. The ...
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0answers
49 views

What is the $\int{w^a(1-w)^b\mathrm d w}$?

I recently came across the $$\int{w^a(1-w)^b\mathrm d w},$$ which looked ridiculously simple at first, but I subsequently discovered that I could not reduce it to some elementary form. To clarify, I ...
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0answers
30 views
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1answer
23 views

Converting a system of first order differential equations to a higher order differential equation

I have read this:Systems of Differential Equations and higher order Differential Equations and I am trying to apply it to the following set of equations but with not much success $$L_{11} \frac{dI_1}{...
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2answers
65 views

Solution of the following differential equation .

$2x^3ydy + (1-y^2)(x^2y^2 -1)dx =0$ The correct solution to the problem is : $x^2y^2 = (Cx - 1)(1 - y^2)$ Where C is a constant .
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2answers
39 views

Differential equation not working out

$$ x'= \frac{-x}{t}$$ So I'm trying to solve the above equation in the following method. We can write $x'= \frac{dx}{dt}$. this makes $$ \frac{dx}{dt} = \frac{-x}{t} $$ $$ \frac{1}{x}dx = \frac{-1}{t}...
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1answer
38 views

Interest Modeling Problem Mistake?

Adriana opens a savings account with an initial deposit of $\$1000$. The annual rate is $6\%$, compounded continuously. Adriana pledges that each year, her annual deposit (deposited continuously) will ...
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1answer
20 views

This D. E. is a homogeneous differential equation?

I'm confused with this differential equation. I read this in a post and in an example in a book. Cite: The best and the simplest test for checking the homogeneity of a differential equation is as ...
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2answers
27 views

Differential equation with no solution that satisfies a certain condition

It is my first week dealing with Differential Equations, and I am stuck at the following question. I am not sure how to approach this, and any help would be greatly appreciated. Let $L$ be a positive ...
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0answers
12 views

Theoretical question related to a differential equation, with attached initial value problem

It is my first week dealing with Differential Equations, and I am lost at the following question: Show that the equation $P(x,y)dx+Q(x,y)dy=0$ has an integrating factor of type $μ\frac{y}{x}$ if and ...
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1answer
31 views

Solving a differential equation based on integrals [on hold]

It is my first week dealing with Differential Equations, and I am totally lost at solving the following equation: $\int^x_0(x-t)y(t)dt=2x+\int^x_0y(t)dt$ Any help would be greatly appreciated!